solnshw08 - 1 Solution to Problem 7.24(a Let X =(X1 Xn...

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1 Solution to Problem 7.24 (a) Let X = ( X 1 , . . . , X n ) denote the data vector. The posterior density is π ( λ | X = x , α, β ) = m ( x ) - 1 bracketleftBigg productdisplay i λ x i x i ! e - λ bracketrightBigg λ α - 1 Γ( α ) β α e - λ/β , λ > 0 , = C ( x , α, β ) λ i x i + α - 1 e - ( n +1 ) λ , λ > 0 , where m ( x ) is the marginal PMF for x after integrating out λ , that is m ( x ) = integraldisplay 0 bracketleftBigg productdisplay i λ x i x i ! e - λ bracketrightBigg λ α - 1 Γ( α ) β α e - λ/β dλ. However, we don’t have to do any calculations here. In the second expression for the posterior PDF, C ( x , α, β ) stands for something that doesn’t depend on λ ; whatever it is, it will make π ( λ | X = x , α, β ) integrate to 1 (integrting over λ ). We see that the form of the PDF is a power of λ times a (negative) exponential in λ (and of course, λ > 0). This can only be a gamma PDF, and we can read off the parameters: the posterior distribution is gamma ( i x i + α, ( n + 1 ) - 1 ). (b) Using the formulae for the mean and variance of the gamma distribution, we have E [ λ | X = x , α, β ] = i x i + α n + 1 Var[ λ | X = x , α, β ] = i x i + α ( n + 1 ) 2 . Note that for large n , the posterior mean is close to the sample mean (which would be the estimator a frequentist would use; it is the MLE and UMVUE). The MLE of the variance of the ¯ X would be ¯ X/n (since the variance of ¯ X is λ/n , and this is close to the posterior

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